Remark 3.13 (pre-Fredholm modules vs. Fredholm modules) 
According to [1]}, a
genuine Fredholm module is defined by a self-adjoint involution
\(F=F^*=F^{-1}\) . In our case the involution property is violated since the
difference \(F^2_{B,\varepsilon }-{\bf 1}\)  is \(\varepsilon \)  times a compact operator. On the other
hand the presence of \(\varepsilon \)  is needed to make the operator \(D_B\)  invertible. The
violation of the involutive property is not a big issue. Indeed, as discussed
in [1]} there is a canonical procedure to
associate a genuine Fredholm module to a given pre-Fredholm module. Another
possibility is to define the \(\varepsilon =0\)  regularization of the resolvent of
\(D_B\)  as described in [3]} or [1]} Let \(V_0\)  be the one-dimensional kernel of \(D_B\)  and \(P_{V_0}\)  the orthogonal projection on \(V_0\) . Then, one can define
\(|D_{B,0}|^{-1}:=|D_B|^{-1}({\bf 1}-P_{V_0})\)  on the orthogonal complement
\(V_0^\bot \)  (where the inverse of \(|D_B|\)  is well defined). One introduces the
extended Hilbert space \(\widehat{{H}}_4:=V_0^\bot \oplus (V_0\oplus V_0)\)  and with the
identification \(V_0\oplus V_0\simeq 2\) , one defines
\(F_{B,0} :=D_B|D_{B,0}|^{-1} \oplus \left(\begin{matrix}0 & 1 \\1 & 0\end{matrix}\right) .\) 
